Approaches to Quantum Gravity

452 C. Burgesswith the following scalar potentialV = λ2 (φ ∗ φ − v 2) 2. (23.2)4This theory is renormalizable, so we can compute its quantum implications in somedetail.Since we return to it below, it is worth elaborating briefly on the criterion forrenormalizability. To this end we follow standard practice and define the ‘engineering’dimension of a coupling as p, where the coupling is written as (mass) p inunits where = c = 1 (which are used throughout). 1 For instance the couplingλ 2 which pre-multiplies (φ ∗ φ) 2 above is dimensionless in these units, and so hasp = 0, while the coupling λ 2 v 2 pre-multiplying φ ∗ φ has p = 2.A theory is renormalizable if p ≥ 0 for all of its couplings, and if for any givendimension all possible couplings have been included consistent with the symmetriesof the theory. Both of these are clearly true for the Lagrangian of eqs. (23.1)and (23.2), since all possible terms are written consistent with p ≥ 0 and the U(1)symmetry φ → e iω φ.23.2.1.1 Spectrum and scatteringWe next analyze the spectrum and interactions, within the semiclassical approximationwhich applies in the limit λ ≪ 1. In this case the field takes a nonzeroexpectation value, 〈φ〉 =v, in the vacuum. The particle spectrum about this vacuumconsists of two weakly-interacting particle types, one of which – ϕ 0 –ismassless and the other – ϕ m –hasmassm = λv. These particles interact withone another through an interaction potential of the formV = 1 [mϕ m +λ ( ) ] 22 2 √ ϕm 22+ ϕ2 0 , (23.3)as may be seen by writing φ = v + (ϕ m + iϕ 0 )/ √ 2. For instance, these interactionsimply the following invariant scattering amplitude for the scattering processϕ 0 (p)ϕ 0 (q) → ϕ 0 (p ′ )ϕ 0 (q ′ ):( )A =− 3λ2 λm 2 [2 + 1√2(p + q) 2 + m 2 − iɛ+1(p − p ′ ) 2 + m 2 − iɛ + 1(p − q ′ ) 2 + m 2 − iɛ]. (23.4)This amplitude has an interesting property in the limit that the centre-of-massscattering energy, E, is much smaller than the mass m. As may be explored by1 It is implicit in this statement that the relevant fields are canonically normalized, and so have dimensionlesskinetic terms.